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Chapter 3: Problem 18

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$P(x)=\frac{1}{4}(x+1)^{3}(x-3)$$

Short Answer

Expert verified
The graph has x-intercepts at \(x = -1\) and \(x = 3\), a y-intercept at \((0, -\frac{3}{4})\), rises as \(x \to \pm \infty\), and turns at \(x = -1\).

Step by step solution

01

Identify the Leading Term and Degree

To determine the end behavior of the polynomial function, identify the leading term. For this we multiply the highest degree terms in each factor. The leading term is found by expanding \((x+1)^3(x-3)\). The highest degree terms are \(x^3\) and \(x\), so multiplication gives \((x^3)(x) = x^4\). Thus, the polynomial is of degree 4 with a leading term \(\frac{1}{4}x^4\).
02

Analyze the End Behavior

The leading term \(\frac{1}{4}x^4\) informs us about the end behavior of the polynomial. Since the coefficient \(\frac{1}{4}\) is positive and the degree is even (4), both ends of the polynomial will rise as \(x\) approaches \(\pm \infty\).
03

Find the x-intercepts

The x-intercepts occur where \(P(x) = 0\). Set \((x+1)^3(x-3)=0\). This gives the x-intercepts at \(x=-1\) and \(x=3\). Notice that \(x=-1\) is of multiplicity 3, resulting in a behavior where the graph touches and turns around at the intercept.
04

Find the y-intercept

To find the y-intercept, substitute \(x=0\) into \(P(x)\). This gives \(P(0) = \frac{1}{4}(0+1)^3(0-3) = \frac{1}{4}(1)(-3) = -\frac{3}{4}\). So, the y-intercept is \(0, -\frac{3}{4}\).
05

Sketch the Graph

Plot the intercepts found. Mark the x-intercepts at \((-1, 0)\) and \((3, 0)\) with \(x=-1\) being a point where the graph turns due to multiplicity 3. Mark the y-intercept at \(\left(0, -\frac{3}{4}\right)\). Ensure that as \(x \to \pm \infty\) the graph rises, respecting the polynomial's end behavior. The graph should touch and turn at \(x = -1\) and pass through \(x = 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

End Behavior
Understanding end behavior means knowing what happens to the graph of a polynomial as the values of \(x\) become very large or very small. To find the end behavior, we look at the leading term of the polynomial, which is the term with the highest degree. In the polynomial function \(P(x)=\frac{1}{4}(x+1)^{3}(x-3)\), the leading term is determined by multiplying the highest degree terms from each factor, providing us \(x^4\).The leading term is \(\frac{1}{4}x^4\). The coefficient \(\frac{1}{4}\) is positive and since the degree is even, both ends of the graph will rise. This means as \(x\) approaches infinity or negative infinity, the graph goes upwards.
X-intercepts
The x-intercepts of a graph are the points where the graph crosses the x-axis. This happens where \(P(x) = 0\). For our polynomial \((x+1)^3(x-3) = 0\), set each factor equal to zero to find the x-intercepts.
  • The factor \((x+1)^3=0\) results in an intercept at \(x=-1\).
  • The factor \((x-3)=0\) results in an intercept at \(x=3\).
Thus, the x-intercepts are \(x=-1\) and \(x=3\). These points are important for understanding the overall shape of the graph.
Y-intercept
The y-intercept is the point where the graph crosses the y-axis. To find the y-intercept, we set \(x = 0\) in the polynomial and find \(P(x)\). For the function \(P(x)=\frac{1}{4}(x+1)^{3}(x-3)\), substituting \(x=0\) gives:\[P(0) = \frac{1}{4}(0+1)^3(0-3) = \frac{1}{4}(1)(-3) = -\frac{3}{4}\]This calculation shows us that the y-intercept is at \(\left(0, -\frac{3}{4}\right)\). Note that this point will always be alone since a polynomial can only cross the y-axis once.
Multiplicity
In polynomial graphing, multiplicity refers to how many times a particular root repeats. For example, in the polynomial \(P(x)=\frac{1}{4}(x+1)^{3}(x-3)\), the factor \((x+1)^3\) has a root at \(x=-1\) with multiplicity 3.
  • A root with an odd multiplicity, like 3, causes the graph to touch and turn horizontally at the x-axis.
  • If the multiplicity were even, the graph would simply touch the x-axis and go back in the same direction without crossing.
The other root \(x = 3\) has a multiplicity of 1, allowing the graph to pass straight through this x-axis point. Understanding these behaviors helps us correctly sketch and visualize the graph's movement at these intercepts.

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Most popular questions from this chapter

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. $$f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, g(x)=1-x^{2}$$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Rule of Signs, the quadratic formula, or other factoring techniques. $$P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6$$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$P(x)=8 x^{3}+10 x^{2}-39 x+9 ; \quad a=-3, b=2$$

The quadratic formula can be used to solve any quadratic (or second-degree) equation. You may have wondered if similar formulas exist for cubic (third- degree), quartic (fourth-degree), and higher-degree equations. For the depressed cubic \(x^{3}+p x+q=0\) Cardano (page 296 ) found the following formula for one solution: $$x=\sqrt[3]{\frac{-q}{2}+\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}+\sqrt[3]{\frac{-q}{2}-\sqrt{\frac{q^{2}}{4}+\frac{p^{3}}{27}}}$$ A formula for quartic equations was discovered by the Italian mathematician Ferrari in \(1540 .\) In 1824 the Norwegian mathematician Niels Henrik Abel proved that it is impossible to write a quintic formula, that is, a formula for fifth-degree equations. Finally, Galois (page 273 ) gave a criterion for determining which equations can be solved by a formula involving radicals. Use the cubic formula to find a solution for the following equations. Then solve the equations using the methods you learned in this section. Which method is easier? (a) \(x^{3}-3 x+2=0\) (b) \(x^{3}-27 x-54=0\) (c) \(x^{3}+3 x+4=0\)

Graph the rational function and find all vertical asymptotes, \(x\)- and \(y\)-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$y=\frac{x^{4}}{x^{2}-2}$$
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